Proceedings of the American Mathematical Society

Author(s): Martin Bohner; Ondrej Doslý
Journal: Proc. Amer. Math. Soc. 129 (2001), 2715-2726.
MSC (2000): Primary 39A12; Secondary 39A11, 34K11
Posted: February 15, 2001
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The classical Prüfer transformation has proved to be a useful tool in the study of Sturm-Liouville theory. In this paper we introduce the Prüfer transformation for self-adjoint difference equations and use it to obtain oscillation criteria and other results. We then offer an extension of this approach to the case of general symplectic systems on time scales. Time scales have been introduced in order to unify discrete and continuous analysis, and indeed our results cover as special cases both the Prüfer transformation for differential and for difference equations.

1.: R. P. Agarwal.
Difference Equations and Inequalities.
Marcel Dekker, Inc., New York, 1992. MR 92m:39002
2.: R. P. Agarwal and M. Bohner.
Basic calculus on time scales and some of its applications.
Results Math., 35(1-2):3-22, 1999. MR 99m:41030
3.: C. D. Ahlbrandt and A. C. Peterson.
Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations.
Kluwer Academic Publishers, Boston, 1996.
4.: D. Anderson.
Discrete trigonometric matrix functions.
Panamer. Math. J., 7(1):39-54, 1997. MR 99e:39004
5.: B. Aulbach and S. Hilger.
Linear dynamic processes with inhomogeneous time scale.
In Nonlinear Dynamics and Quantum Dynamical Systems. Akademie Verlag, Berlin, 1990. MR 91g:58266
6.: J. H. Barrett.
A Prüfer transformation for matrix differential equations.
Proc. Amer. Math. Soc., 8:510-518, 1957. MR 19:415f
7.: M. Bohner and O. Doslý.
Disconjugacy and transformations for symplectic systems.
Rocky Mountain J. Math., 27(3):707-743, 1997. MR 99e:39007
8.: M. Bohner and O. Doslý.
Trigonometric transformations of symplectic difference systems.
J. Differential Equations, 163:113-129, 2000. CMP 2000:11
9.: O. Doslý.
On some properties of trigonometric matrices.
Cas. Pest. Mat., 112:188-196, 1987. MR 88g:34041
10.: O. Doslý and R. Hilscher.
Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales,
2000, J. Differ. Equations Appl., to appear.
11.: Á. Elbert.
A half-linear second order differential equation.
Colloq. Math. Soc. János Bolyai, 30:158-180, 1979. MR 84g:34008
12.: L. Erbe and S. Hilger.
Sturmian theory on measure chains.
Differential Equations Dynam. Systems, 1(3):223-244, 1993. MR 95b:39002
13.: G. J. Etgen.
A note on trigonometric matrices.
Proc. Amer. Math. Soc., 17:1226-1232, 1966. MR 35:4504
14.: G. J. Etgen.
Oscillation properties of certain nonlinear matrix equations of second order.
Trans. Amer. Math. Soc., 122:289-310, 1966.
15.: H. Heuser.
Gewöhnliche Differentialgleichungen.
B. G. Teubner, Stuttgart, 1989. MR 90f:34001
16.: S. Hilger.
Analysis on measure chains - a unified approach to continuous and discrete calculus.
Results Math., 18:18-56, 1990. MR 91m:26027
17.: S. Hilger.
Special functions, Laplace and Fourier transform on measure chains.
Dynam. Systems Appl., 8(3-4):471-488, 1999.
Special Issue on ``Discrete and Continuous Hamiltonian Systems'', edited by R. P. Agarwal and M. Bohner. CMP 2000:04
18.: W. G. Kelley and A. C. Peterson.
Difference Equations: An Introduction with Applications.
Academic Press, San Diego, 1991. MR 93f:39002
19.: H. Prüfer.
Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen.
Math. Ann., 95:499-518, 1926.
20.: W. T. Reid.
A Prüfer transformation for differential systems.
Pacific J. Math., 8:575-584, 1958. MR 20:5913
21.: W. T. Reid.
Generalized polar coordinate transformations for differential systems.
Rocky Mountain J. Math., 1(2):383-406, 1971. MR 43:6488

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 39A12, 39A11, 34K11

Martin Bohner
Affiliation: Department of Mathematics and Statistics, University of Missouri--Rolla, 313 Rolla Building, Rolla, Missouri 65409-0020
Email: bohner@umr.edu

Ondrej Doslý
Affiliation: Mathematical Institute, Czech Academy of Sciences, Zizkova 22, CZ--61662 Brno, Czech Republic
Email: dosly@math.muni.cz

DOI: 10.1090/S0002-9939-01-05833-6
PII: S 0002-9939(01)05833-6
Keywords: Pr\"ufer transformation, Sturm-Liouville difference equations, linear Hamiltonian difference systems, time scales, symplectic systems
Received by editor(s): September 2, 1999
Received by editor(s) in revised form: January 24, 2000
Posted: February 15, 2001
Additional Notes: The research of the first author was supported by the University of Missouri Research Board. The research of the second author was supported by the Grant G201/98/0677 of the Czech Grant Agency (Prague).
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2001, American Mathematical Society

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